The forward-backward envelope (FBE) is an exact, continuous, real-valued merit function for (REF ) [1]}, [2]}, [3]}, [4]}, [5]}. Function \(\ell \)  can be approximated at a point \(u\in U\)  by the quadratic upper bound
\(Q^\ell _\gamma (v; u) = \ell (u) + \nabla \ell (u)^{\scriptscriptstyle \top }(v-u) + {1}{2\gamma }\Vert u-v\Vert ^2.\) 
